Finding k Disjoint Triangles in an Arbitrary Graph

Abstract

We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H = K 3 (the triangle) we give an O(2 2k log k+1.869k n 2 ) algorithm, and for general H an O(2 k|H| log k+2k|H| log |H| n |H| ) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H = K 3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k 3 ) vertices in polynomial time. INTRODUCTIONFor a fixed graph H and an input graph G, the H-packing problem asks for the maximum number of vertex-disjoint copies of H in G. The K 2 -packing (edge packing) problem, which is equivalent to maximum matching, played a central role in the history of classical computational complexity. The first step towards the dichotomy of 'good' (polynomialtime) versus 'presumably-not-good' (NP-hard) was made in a paper on maximum matching from 1965 [E65], which gave a polynomial time algorithm for that problem. On the other hand, the K 3 -packing (triangle packing) problem, which is our main concern in this paper, is NP-hard [HK78].Recently, there has been a growing interest in the area of exact exponential-time algorithms for NP-hard problems. When measuring time in the classical way, simply by the 1

Linear Problem Kernels for NP-Hard Problems on Planar Graphs

“…Given a graph G = (V, E) and a non-negative integer k, the Maximum Triangle Packing problem asks for a set P of at least k vertex-disjoint triangles in G. The set P is called a triangle packing of G. This problem is NP-complete on planar graphs [14]. A problem kernel with O(k 3 ) vertices is known for general graphs [10].…”

A Problem Kernelization for Graph Packing

“…The problem kernel with O(k 3 ) vertices by Fellows et al [9] starts with a greedy packing P of triangles, which contains less than 3k vertices (otherwise, we already have a packing of k triangles). Then, based on the size of P, the number of vertices in V \ V (P) is bounded, which implies that the total number of vertices in the graph is bounded, yielding a problem kernel.…”

“…The problem with this approach is that there is too much structure of the graph "outside" of P. To deal with this problem, we use a different notion of witness, which contains more structure than P, but which is still small enough in order to obtain a better bound on the number of vertices. Our kernelization is based on the same reduction rules as the kernel by Fellows et al [9]. However, our approach applies them differently, and, most importantly, it uses a different analysis.…”

“…We call this algorithm compute witness. After computing T , the following data reduction rule due to Fellows et al [9] is applied.…”

“…For more about kernelization we refer to a recent survey by Guo and Niedermeier [16]. In this work, we improve a bound of 108k 3 −73k 2 −18k vertices for a problem kernel for the problem of packing h vertex-disjoint triangles [9] to a bound of 45k 2 vertices. Moreover, we generalize our approach to a problem kernel of O(k |V (H)|−1 ) vertices for the problem of packing k vertex-disjoint copies of a fixed connected graph H.…”